Finite Group Databasebeta

← #27276726

Subgroups — #27276726

Order 13568.

1026 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274639C212yesmaximal
2#20273045C214yesmaximal
3C22×C4⋊Q8×C53C211yesmaximal, characteristic
4C2×#13133136—24no—
5#16810587C2214yes—
6#16810349—28no—
7C23×Q8×C53C411yescharacteristic
8C23×Q8×C53C2211yescharacteristic
9C2×C4⋊Q8×C53—24no—
10C2×C4⋊Q8×C53—28no—
11C22×C4⋊C4×C53C412yes—
12C22×C4⋊C4×C53C2212yes—
13C22×C42×C53C411yescharacteristic
14C2×#11719917—24no—
15#13133136—48no—
16C22⋊#88322—28no—
17C22×Q8×C53—24no—
18C22×Q8×C53—22no—
19C22×Q8×C53—24no—
20C22×Q8×C53—44no—
21C22×Q8×C53—44no—
22C22×Q8×C53D414yes—
23C23×C4×C53C2×C412yes—
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53C2×C412yes—
27C23×C4×C53C2311yescharacteristic
28C4⋊Q8×C53—232no—
29C2×C4⋊C4×C53—24no—
30C2×C4⋊C4×C53—24no—
31C2×C4⋊C4×C53—24no—
32C2×C4⋊C4×C53—24no—
33C2×C4⋊C4×C53—24no—
34C2×C4⋊C4×C53D418yes—
35C2×C42×C53—22no—
36C2×C42×C53—24no—
37C24×C53C22×C411yescharacteristic
38C2×Q8×C53—28no—
39C2×Q8×C53—28no—
40C2×Q8×C53—28no—
41C2×Q8×C53—44no—
42C2×Q8×C53—416no—
43C2×Q8×C53—416no—
44C2×Q8×C53—44no—
45C2×Q8×C53—44no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—24no—
50C22×C4×C53—28no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53—44no—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22⋊C414yes—
60C22×C4×C53C22⋊C414yes—
61C22×C4×C53C2×D412yes—
62C22×C4×C53C2×D414yes—
63C4⋊C4×C53—216no—
64C4⋊C4×C53—216no—
65C42×C53—28no—
66#11719917—48no—
67C2×#88322—44no—
68C23×C53—22no—
69C23×C53—24no—
70C23×C53C2×C22⋊C411yescharacteristic
71C23×C53C2×C22⋊C412yes—
72Q8×C53—416no—
73Q8×C53—416no—
74C2×C4×C53—28no—
75C2×C4×C53—28no—
76C2×C4×C53—28no—
77C2×C4×C53—22no—
78C2×C4×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53—22no—
83C2×C4×C53—24no—
84C2×C4×C53—24no—
85C2×C4×C53—44no—
86C2×C4×C53—44no—
87C2×C4×C53—44no—
88C2×C4×C53—44no—
89C2×C4×C53—44no—
90C2×C4×C53#9914yes—
91C2×C4×C53#10118yescomplemented
92C2×C4×C53#12014yes—
93#88322—84no—
94#8551—531nomaximal, Sylow
95C22×C53—28no—
96C22×C53—22no—
97C22×C53—22no—
98C22×C53—22no—
99C22×C53#37911yescharacteristic
100C22×C53C2×#9912yes—
101C22×C53C2×#10114yes—
102C4×C53—24no—
103C4×C53—24no—
104C4×C53—28no—
105C4×C53—42no—
106C4×C53—44no—
107C22×C4⋊Q8D5311yescharacteristic
108#1784—532no—
109#1463—534no—
110C2×C53—22no—
111C2×C53—24no—
112C2×C53#178111yescharacteristic
113C2×C53#178412yes—
114C23×Q8Dic5311yescharacteristic
115C23×Q8C2×D5311yescharacteristic
116C2×C4⋊Q8—24no—
117C2×C4⋊Q8—28no—
118C22×C4⋊C4Dic5312yes—
119C22×C4⋊C4C2×D5312yes—
120C22×C42Dic5311yescharacteristic
121C2×#101—1064no—
122#407—534no—
123#324—1068no—
124C53#855111yescharacteristic, complemented, Sylow
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8C22⋊D5314yes—
131C23×C4C2×Dic5312yes—
132C23×C4C2×Dic5311yescharacteristic
133C23×C4C2×Dic5311yescharacteristic
134C23×C4C2×Dic5312yes—
135C23×C4C22×D5311yescharacteristic
136C2×M16—1064no—
137C4⋊Q8—232no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4C22⋊D5318yes—
144C2×C42—22no—
145C2×C42—24no—
146#101—2128no—
147C22⋊C8—1068no—
148C24C22×Dic5311yescharacteristic
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—28no—
152C2×Q8—44no—
153C2×Q8—416no—
154C2×Q8—416no—
155C2×Q8—44no—
156C2×Q8—44no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4C22⋊Dic5312yes—
169C22×C4C22⋊Dic5312yes—
170C22×C4C22⋊Dic5314yes—
171C22×C4C22⋊Dic5314yes—
172C22×C4C2×C22⋊D5312yes—
173C22×C4C2×C22⋊D5314yes—
174M16—2128no—
175C2×C8—2124no—
176C4⋊C4—216no—
177C4⋊C4—216no—
178C42—28no—
179C23—22no—
180C23—24no—
181C23C2×C22⋊Dic5311yescharacteristic
182C23C2×C22⋊Dic5312yes—
183Q8—416no—
184Q8—416no—
185C2×C4—28no—
186C2×C4—28no—
187C2×C4—28no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4#1313313214yes—
202C2×C4#1313313618yescomplemented
203C2×C4#1313318614yes—
204C8—4244no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22#1681047911yescharacteristic
210C22C2×#1313313212yes—
211C22C2×#1313313614yes—
212C4—24no—
213C4—24no—
214C4—28no—
215C4—42no—
216C4—44no—
217C2—22no—
218C2—24no—
219C2#2027463011yescharacteristic
220C2#2027463912yes—
subgroupG/H
centerC22#16810479
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC22×C4⋊Q8×C53C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenterC22×C4⋊Q8D53
hypocenterC53#8551
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
91. C2×C4×C53#101semidirect
124. C53#8551semidirect
202. C2×C4#13133136semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202746392
#202730452
C22×C4⋊Q8×C532
#855153